dx₁ x₁+6x2, x₁(0)=12 dt Given the system of linear DEQS dt dx=5x+2x2 =5x+2x2, x2(0)=1 16 The coefficient matrix M= has eigenvalues ₁ =7, 2₂ =-4. Moreover, a particular eigenvector of 5 2 X1 2₁₁ is X2 30 1 Χ1 -6 and a particular eigenvector of λ is 5 Using the given facts above, do the following: 1. Generate the matrix P using particular eigenvectors of M. 2. Find the inverse matrix P-1 3. Find the diagonal matrix D using the fact that D=P-¹MP. 4. Find the disconnected system using the definition X-PY (write it in matrix form). 5. Write the disconnected system in regular form. 6. Solve each of the DEQs using the separation of variables method to find the most general solutions in terms of the y variables. 7. Write the most general solutions in terms of the x-variables. 8. Find the integration constants using the initial conditions. 9. Write the particular solutions of the system. 10. Check that the particular solutions satisfy the DEQS and the initial conditions.